How do I use my CFV calibration data?

One of the flow meters types we calibrate at CEESI is the Critical Flow Venturi (CFV).  The basic concept of how a CFV can be used to measure flow is that if you have achieved a sufficiently low pressure ratio (exit pressure to inlet pressure) the flow through the CFV will become critical or choked.  Once you have established that the flow is choked you can use the inlet pressure, inlet temperature and some fluid properties to determine the flowrate.  This is the equation for calculating flow through a CFV from ASME MFC 7M.

A quick rundown of all the parts of the equation; qm is the mass flowrate, A* is the area of the throat, R is the universal gas constant, M is the molecular mass of the fluid, P is the inlet pressure, T is the inlet temperature, C* is known as the critical flow function and is a property that can be determined from inlet P&T and fluid.  The last piece of the equation is Cd, discharge coefficient, is the most important to a calibration company like CEESI.  Cd is the value that corrects the equation from theoretical flow rates to real flowrates.  There are ways to estimate Cd in the standards that can be useful, or you could have the CFV calibrated.

Cd is known to vary with Reynolds number so it is best to calibrate your CFV over the full Reynolds number range that it may be used to measure flow.  Now Reynolds number is value that is calculated from the flowrate, viscosity of the fluid and diameter of the throat of the CFV. 

I have seen data from a CFV calibration regressed to a curve fit 2 ways.  The first is Cd vs Reynolds number.  The result of this a fit where you can determine a Cd based on a Reynolds number.  The other regression is kfactor vs pressure.  This kfactor is a simplification of the flow equation for CFVs and is seen below.

You can see an example below of how each fit looks for a single set of calibration results. 

Now that we have covered the background information, we can get to the point of this article.  Why would you want to use this kfactor to calculate flow rate instead of Cd?  Because it simplifies the calculations needed to do the calculations.  There is no need to do property calculations which can be painful to do.  Also, it eliminates the need to iterate.  You may have noticed that qm is in the equation for Reynolds number even though you are using Reynolds number to determine mass flowrate.  This forces the user to have to iterate the calculation of mass flow and Reynolds number until the values stop changing.  The kfactor equation does not contain a circular calculation like this and can be completed with only the measured pressure and temperature.

Now there are some disadvantages to the kfactor method as well.  The reason it works at all is that there are some assumptions made that allow the simplification of the equation.  First is that fluid is the same at calibration as when the meter is being used.  In other words, you cannot change fluids and use the same values.  Secondly the method assumes that the temperature of the fluid during use is close to the calibration temperature.  This allows the errors caused by ignoring critical flow factor and Reynolds number effects to be small.

The plot below shows how the trendline of how the errors add up when the temperature deviates from the calibration temperature based on the example calibration above.  The slope of this line partly depends on how much slope there is in the Cd vs Reynolds number curve.

In conclusion, Kfactor can be an acceptable way to simplify CFV flow rate calculations, but there are drawbacks.